Broadie, M., Glasserman, P. & Kou, S.G. (1999) Connecting Discrete and Continuous Path-Dependent Options Finance Stochast. 3, 55-82  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... its embedded Gaussian random walk was studied in [4] Correction terms for the di#usion approximation to one and two barrier crossing problems were examined in [29] In the context of option pricing, a corresponding relationship between the continuous and discrete time models was investigated in [8, 9]. 9 4.2 Approximation of the delay probability # As seen from Corollary 1, the probability of wait is expressed in terms of an infinite sum of Gaussian functions. While the sum converges fast for moderately large # # (say # # 1) it has particularly slow rate of convergence for small values ....

M. Broadie, P. Glasserman, and S. Kou. Connecting discrete and continuous path-dependent options. Finance and Stochastics, 3:55--82, 1999.

.... can be computationally expensive without the enhancement of variance reduction techniques and one must account for the inherent discretization bias resulting from the approximation of continuous time processes through discrete sampling (see Broadie and Glasserman [3] Broadie, Glasserman and Kou [4] and Kemma and Vorst [12] This work was supported by the National Science Foundation under grant DMS 98 02464. y I would like to thank Fredrik Akesson, Julien Hugonnier, Steven Shreve, Dennis Wong and Mingxin Xu for helpful comments and suggestions on this paper. 1 In general, the price ....

Broadie, M., Glasserman, P., Kou, S. \Connecting discrete and continuous path-dependent options", Finance and Stochastics, 3, 1999, 55-82. 8

....called fixed strike look back options, have payoff which depends only on the distribution of the high or low, for example (H Gamma K) in the case of a hindsight call option. There is a large number of papers devoted to the valuation of such options. For details, see the references in Broadie, Glasserman and Kou (1996). In this paper, we will identify some of the simple properties of the joint distribution of the high, low, close which are particularly useful when simulating the value of look back and barrier options without simulating the sample paths of the process. The diffusions considered are all related ....

....where X has the distribution of DeltaX = X( j 1) Deltat) Gamma X(j Deltat) and using the corresponding continuous time result. Such a crude adjustment improves the approximation, although more refined adjustments have been suggested in the literature (cf. for example Levy and Mantion(1996) Broadie, Glasserman and Kou (1996)) In the latter paper, using results from Asmussen, Glynn and Pitman (1995) it is observed that the expected difference between the high H for a continuously observed process and the high H Delta for an identical process monitored at discrete time intervals of length Delta takes the form fi 1 ....

Broadie, M. Glasserman, P. and Kou, S. (1996) Connecting Discrete and Continuous Path-dependent Options. preprint.

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Broadie, M., Glasserman, P. & Kou, S.G. (1999) Connecting Discrete and Continuous Path-Dependent Options Finance Stochast. 3, 55-82

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Broadie, M, Glasserman, P. and Kou, S. G. (1999). Connecting discrete and continuous path-dependent options. Finance and Stochastics.3 ,55--82.

....the barrier. Unfortunately, the exact pricing results available for continuous barriers do not extend to the discrete case. 1 Even numerical methods using standard lattice techniques or Monte Carlo simulation face significant difficulties in incorporating discrete monitoring, as demonstrated in Broadie, Glasserman, and Kou (1996). In this paper, we introduce a simple continuity correction for approximate pricing of discrete barrier options. Our method uses formulas for the prices of continuous barrier options but shifts the barrier to correct for discrete monitoring. The shift is determined solely by the monitoring ....

....are down and out call options, i.e. S 0 H , and the option becomes worthless if S crosses the barrier before the option expires at time T . The true value is determined from a trinomial procedure modified in several ways to specifically handle discrete barriers and is fully described in Broadie et al. 1996). This numerical procedure has an average accuracy of about 0.001 for this range of parameters. Table 2.1 and later tables show that the continuous barrier price can differ from the discrete barrier price by economically significant amounts. For example, for H = 97 in Table 2.1, the continuous ....

BROADIE, M., P. GLASSERMAN, and S. KOU (1996): "Connecting Discrete and Continuous PathDependent Options," working paper, Columbia Business School, New York, to appear in Finance and Stochastics.

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Broadie, M., Glasserman, P. and and Kou, S.G. (1999) Connecting discrete and continuous path-dependent options. Finance and Stochastics 3, 55--82.

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Broadie, M., Glasserman, P. & Kou, S.G. (1999) Connecting Discrete and Continuous Path-Dependent Options Finance Stochast. 3, 55-82

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Broadie, M., P. Glasserman, and S. Kou (1999), "Connecting Discrete and Continuous Path-Dependent Options," Finance and Stochastics, Vol.3, 55-82.

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Broadie, M., Glasserman, P. and S. Kou, 1999, Connecting Discrete and Continuous PathDependent Options, Finance and Stochastics 3, 55-82.

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